Coherent sheaves on subvarieties in Hopf manifolds

Abstract

We prove a version of GAGA theorem for a normal complex analytic variety $X$ equipped with an invertible holomorphic contraction $\gamma$ with center in $x$. We show that $X$ admits a natural structure of an affine variety, and any $\gamma$-equivariant complex analytic reflexive coherent sheaf on $X$ admits a natural algebraic structure. We prove a structure theorem for $X_0:=X\backslash x$, showing that it admits a proper action of ${\Bbb C}^*$, and is isomorphic to the space of non-zero vectors in the total space of an ample line bundle over the projective variety $Z:= X_0/{\mathbb C}^*$ equipped with an orbifold structure. We show that the quotient $M:=X_0/\gamma$ admits a holomorphic embedding to a Hopf manifold, and, conversely, any normal subvariety $M$ in a Hopf manifold is obtained this way. We prove a form of structure theorem, showing that any reflexive coherent sheaf on $M$, $\dim M > 2$, admits a filtration such that its associated graded subquotients, tensored with an appropriate line bundle, are obtained as pullbacks of coherent sheaves on the projective variety $Z=X_0/{\mathbb C}^*$. This is used to show that any reflexive coherent sheaf on $M$ is filtrable, that is, admits a filtration with associated graded quotients of rank $\leq 1$.